Basic Principles: Relativity, Maxwell’s Equation’s, and
Accelerator Coordinate Systems
Lecture 1b
S. Cousineau, J. Holmes, R. Potts, Y. Zhang
USPAS
January, 2014
Basic Units and Relationships
Energy and momentum in accelerators are usually expressed in units
of “electron Volts”:
1 eV = 1.602 x 10-19 Joules
We will use energy units:
keV = 103 eV
MeV = 106 eV
GeV = 109 eV
TeV = 1012 eV
Similarly, the units of momentum, p, are eV/c.
In practice, we will sometimes drop the factor of c.
And finally, for mass, the units are eV/c2. For instance
mp = mass proton = 938 MeV/c2
me = mass electron = 511 keV/c2
Relativistic Relationship
In most accelerators, particles move at relativistic speeds, and therefore
we need to use relativistic mechanics to describe particle motion and
fields.
Einstein’s Special Theory of Relativity:
1) The laws of physics apply in all inertial (non-accelerating) reference
frames.
2) The speed of light in vacuum is the same for all inertial observers.
Notice that (1) does not mean that the answer to a physics calculation is
the same in all inertial reference frames. It only means that the physics
law’s governing the calculation are the same.
Example: Consider a light bulb hanging in a boxcar moving at relativistic
velocity. How long does it take a light ray, moving directly down in the boxcar
frame, from the bulb to reach the floor:
a) as computed by an observer in the car?
b) as computed by an observer on the ground?
The answers differ by a factor of:
h
v
(**Calculation**)
The Relativistic Factor
cv
where,
1
12
Therefore time is dilated for the observer on the ground, compared
with the observer in the boxcar.
h
Dt* =h
c
vlight=c
d = vboxcarDt
l = h2 + vboxcarDt( )2
Dt =h2 + vboxcarDt( )
2
c
Dt =h
c
1
1-v2
c2
Dt = Dt*1
1-v2
c2
= gDt* g º1
1-v2
c2
=1
1- b 2 where, b º
v
c £ 1
Boxcar CM frame: Lab frame:
Other Relativistic Relationships
These principles give rise to time dilation and length contraction:
t = t*
L = L*/
The LHS quantities are given in the rest frame of the observer who
perceives an object in motion. We often call this the “lab frame”. The
RHS quantities (*) are in the rest frame of the moving object, often
called the “center of mass” frame.
Time dilation is an important concept in particle physics because many
particles have limited lifetimes. Time dilation says that the particle
lifetimes are longer in the “Lab frame”.
omm
For an observer in the lab frame, the mass of an object also appears to
increase at high velocity. The object becomes infinitely heavy as it
approaches the speed of light.
Relativistic Energy Equations
The factors and are
commonplace in most
relativistic equations:
In fact, the total energy of a
particle (sum of kinetic and rest
energy), is given by:
For accelerators, it is often
convenient to find using the kinetic
energy, T, of a particle:
c
v 2
2
2 1
1
1
1
c
v
E 2 = p2 +moc2
2
2 1 )1(cm
TcmT
o
o
And finally, for the relationship
between momentum and energy,
we have:
E = gmoc2
p = gmov =
222 cmTcmmcE oo
Example Problem
A pion of rest mass mo = 139.6 MeV decays into to another particle in a
time t = 26e-9 seconds, as measured in the pion’s own rest frame. For a
pion that is accelerated to a kinetic energy of T=100 MeV, calculate:
a.The relativistic factors β and γ.
b.The distance the pion will travel in the lab frame before decay.
g =1+T
m0
=1+100
139.6=1.72 b = 1-
1
g 2= 0.81a.
b. t =gt* =1.72*26´10-9 = 4.5´10-8s
Relativistic Beta Function
The function is the speed of a particle divided by the speed of light. As a
massive particle is accelerated, increases asymptotically towards 1 (speed
of light), but never gets there:
• Heavier particles become relativistic at higher energies.
• No particle with finite mass can travel at the speed of light in vacuum (=1). Massless particles always satisfy =1.
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000
Kinetic energy (MeV)B
eta
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000
Kinetic energy (keV)
Be
ta
Electron Proton
Maxwell’s Equations
In accelerators, we use electric fields to accelerate particles and
magnetic fields to guide and focus particles. The standard equations
used to describe the fields are Maxwell’s equations (in MKS units):
Ñ·eE =r
e 0
Ñ·B= 0
e = ereo, eo = permittivity of free space
m = mrmo, mo = permeability of free space
e0moc2 =1
Bt
E
Ñ´mB = m0J +1
c2
¶
¶teE
(For vacuum or
“well-behaved” materials)
A Closer Look
+
Gauss Law for Electric Fields: The total electric field flux
through a surface is equal to the charge enclosed by the
surface (to within a multiplicative factor).
E
Divergence theorem: The divergence integrated
over the volume of a region is equal to the flux
through the surface area of the region.
A Closer Look
0 B
There are no magnetic
monopoles!
Magnetic fields do not
diverge. Net magnetic flux
through a closed surface is
zero.
Magnetic fields lines for a dipole run from North to South. For a field
generated by a current, I, point your right thumb in the direction of current –
your fingers will curl in the direction of B.
I B
A Closer Look
A changing magnetic field
induces an electric field...
A changing electric field
induces a magnetic field…
This concept is important in RF
acceleration of particles.
A Closer Look
Stokes Theorem: Curl integrated over an area inside a closed curve
equals the line integral around the curve.
The “curl” of a vector function is a measure of its “swirl” or “twist”. For the
total “swirl”, all contributions cancel except those at the boundary.
)0/ (if tEIdlB
enclosed
loop r
(Ñ´ò V )·dA dlV
“Stokes’ Law for Magnetic Fields”: For a constant E field, the component of
the B field along any closed path is equal to the total current enclosed.
Scalar Potential
For any material-free field region, if the integral from point A to point B is
independent of the path, then the field can be expressed as the gradient
of a scalar potential.
A B
path 1
path 2
Field Region
So, for electric and magnetic fields in a material-free region, we can write:
BPathA Path
ds (Field)ds (Field)
zdz
dVy
dy
dVx
dx
dVV
VField
ˆˆˆ
B
E
VB
VE
We will find these expressions useful!
The Lorentz Force Equation
For a charged particle passing through an E or B field the force is
governed by the Lorentz Force Equation:
F = q(E +v ´ B)
Force from the
electric field is
in the direction
of E
Force from the magnetic field is
perpendicular to the direction of
v and B, as given by the “Right
Hand Rule”
Right Hand Rule for a=b x c : Point your fingers in the direction of b, then
curl your fingers toward the direction of c, and then your thumb will point in
the direction of a. (**Example**)
dt
pdamF
A force is the change
in momentum with
respect to time.
Lorentz Transformation of Fields
Do the fields E and B look the same in all inertial reference frames?
Example: A particle is passing by an observer at velocity v.
In the “lab frame”, the moving charged
particle produces a current, and thus it has
both an E field and a B field.
But, in the frame of reference moving with
the particle, the particle is at rest and has
only an E field.
B
+
E
v
The Special Theory of Relativity states that the laws of physics, i.e.,
Maxwell’s equations in this case, are the same in all inertial reference
frames. But the results of the laws can appear different in different
reference frames.
+
E
v=0
Lorentz Transformation of the Fields
ss
xsyy
ysxx
EE
BEE
BEE
*
*
*
)(
)(
ss
xsyy
ysxx
BB
EBB
EBB
*
*
*
)(
)(
The transverse fields, E and B, transform according to the following
equations.
Here, the (*) quantities on the left hand side are taken in the reference
frame moving with velocity s, relative to the non-(*) quantities, which are
in the lab frame.
Accelerator Coordinate Systems
In general, any accelerator will be designed (shaped) to give a “reference
trajectory” for particle travel. This reference trajectory is defined by the
physical centers of the beam line elements.
In beam physics, we are generally interested in deviations from the reference
trajectory. Therefore it is most convenient to place the coordinate system
origin on the reference trajectory, and align one (the longitudinal) coordinate
axis with the reference trajectory. The remaining (transverse) axes are chosen
perpendicular to the longitudinal axis.
Longitudinal axis points in the
direction of the reference
trajectory at any point (tangent
to the reference path).
Curvilinear Coordinate System (continued)
• The z (or s) axis of the coordinate system is the instantaneous tangent to the
reference curve.
• Looking down along the z axis, positive x is to the left and in the plane of reference,
and positive y is up and perpendicular to the plane of reference.
uy
uz ux
reference trajectory
ro
actual
trajectory
dr
r
Curvilinear Coordinate System (continued)
uy
uz ux
reference trajectory
ro
actual
trajectory
dr
r r(x, y, z) = r0(z)+ x(z)ux (z)+ y(z)uy(z)
dr(x, y, z)
dz=dr0(z)
dz+dx(z)
dzux (z)+
dy(z)
dzuy(z)+ x(z)
dux (z)
dz+ y(z)
duy(z)
dz
dr0
dz= uz
dux
dz= k0 x
dr0
dz= k0 xuz where k0 x = curvature in x
duy
dz= k0 y
dr0
dz= k0 yuz where k0 y = curvature in y
dr = dzuz (1+k0xx(z)+k0yy(z))
ºh
+dxux +dyuy
dr = hdzuz +dxux +dyuy
Differentiating:
Grouping Terms:
The result is the position relative to an moving origin on the reference trajectory:
The position of a particle w.r.t. a fixed origin:
(See Wiedemann 1.3.3)